[lamode]

  How audible is that?

 
A frequency response dip of 0.1dB is basically inaudible, but it's not the FR anomaly which is significant here.
I wish we could test all of this out properly in the lab. Very interesting stuff.

 

[bigshot]

Do you know of any equipment that has audible TIM? I have never encountered any- all of my equipment is transparent. Are there any reviews of equipment that claim bad TIM ratings? I googled for an hour and couldn't find a single one. It really seems like this is a problem that no longer exists, or perhaps a problem that never existed, like jitter. I suspect that it was an issue with LP playback, which had cartridges capable of very high frequency response and huge transient clicks and pops. In digital audio, particularly redbook playback, I bet it just doesn't exist.

 

[stv014]

  That'll be really obvious though - if the negative feedback isn't 180 degrees out of phase with the input signal at all audio frequencies, your frequency response will be screwed up.

 
It is actually not a problem if there is some phase shift in the negative feedback loop. A typical amplifier has ~90 degree open loop phase shift over most of the audio band, and is often designed to reach 135 degrees at the frequency where the loop gain crosses 0 dB. For more information, read about Bode plot and phase margin. It is not intuitive, and the math is complex (literally), but this topic has been researched extensively decades ago, and having a first order lowpass filter in a negative feedback loop (like it is usually there for compensation purposes) can indeed be perfectly fine, and not only with a sinusoid input.
 
By the way, the formula for calculating the closed loop gain (Afb) from the open loop gain (Aol) and the feedback factor (beta = 1 / intended closed loop gain) is Afb = Aol / (beta * Aol + 1). The gain values are complex numbers. For example, if an amplifier has 60 dB open loop gain at 20 kHz (GBWP = 20 MHz) with a phase margin of 90 degrees, and the feedback loop is designed for 5x gain, then, using the above formula, the real gain will only differ from that by less than 0.0002 dB, with a phase error of less than 0.3 degrees.

 

[cjl]

   
It is actually not a problem if there is some phase shift in the negative feedback loop. A typical amplifier has ~90 degree open loop phase shift over most of the audio band, and is often designed to reach 135 degrees at the frequency where the loop gain crosses 0 dB. For more information, read about Bode plot and phase margin. It is not intuitive, and the math is complex (literally), but this topic has been researched extensively decades ago, and having a first order lowpass filter in a negative feedback loop (like it is usually there for compensation purposes) can indeed be perfectly fine, and not only with a sinusoid input.
 
By the way, the formula for calculating the closed loop gain (Afb) from the open loop gain (Aol) and the feedback factor (beta = 1 / intended closed loop gain) is Afb = Aol / (beta * Aol + 1). The gain values are complex numbers. For example, if an amplifier has 60 dB open loop gain at 20 kHz (GBWP = 20 MHz) with a phase margin of 90 degrees, and the feedback loop is designed for 5x gain, then, using the above formula, the real gain will only differ from that by less than 0.0002 dB, with a phase error of less than 0.3 degrees.

That's what I get for firing off a quick response without much thought. I'm quite familiar with bode plots and phase margin, I just clearly wasn't thinking...
 
Thanks for the correction though.

 

[stv014]

In fact, my reply was also in response to lamode's original claim from post #42 that negative feedback does not "work" (or it only does with a steady sine wave input) if there is any phase shift in the loop. Phase shift by itself is entirely linear and does not add any non-linear distortion, and, as shown above, the feedback can correct it as long as it is within reasonable limits.

 

[lamode]

  In fact, my reply was also in response to lamode's original claim from post #42 that negative feedback does not "work" (or it only does with a steady sine wave input) if there is any phase shift in the loop. Phase shift by itself is entirely linear and does not add any non-linear distortion, and, as shown above, the feedback can correct it as long as it is within reasonable limits.

 
And I appreciated your reply, though I haven't had time to check out the links you provided, unfortunately.

 

[stv014]

I created a simple DSP simulation of a negative feedback. It consists of the following:
- the difference of the input and the feedback signal is multiplied by 100000 (100 dB DC gain)
- it is then passed through a 6 dB/octave "compensation" lowpass filter with a -3 dB corner frequency of 10 Hz (GBWP = 1 MHz)
- the signal is then distorted by the arc tangent function 5 * atan(x / 5) to add some open loop non-linearity
- it is lowpass filtered again with a corner frequency of 1 MHz so that the phase margin decreases somewhat at the unity gain frequency
- the feedback signal for the next sample is calculated by dividing the output by 2 (closed loop gain = 6 dB)
The model was run at a sample rate of 10 MHz, so there is a 1 sample (100 ns) delay in the feedback loop. Although this could be considered a basic simulation of additional delays in the system.
 
On the following waveform and FFT displays, the left (top, cyan) channel is the unmodified input signal, and the right channel (bottom, magenta) is the output of the simulated feedback, divided by 2 to remove the gain. Time is shown in samples, 1 sample is 100 ns.
 
-6 dBFS step response (the ripples on the left channel were added by the waveform display):
   
 
The same step with a 10 kHz -12 dBFS sine wave added, the FFT is from some time after the step to show the distortion:
   
 
The first few cycles of the 19+20 kHz IMD test, and the distortion spectrum (again after some delay, otherwise it would just show the spectrum of the transient):
   
 
Of course, this is only a simplified model, but it does show that a negative feedback loop with a typical compensation filter added can handle transients in the audio band without noticeable problems, such as needing a few cycles to "stabilize" after a tone is turned on, or causing severe phase errors. The results from recording the output of a real amplifier would be similar, and links to difference extraction tests with music have also been posted earlier.

 

[lamode]

  I created a simple DSP simulation of a negative feedback. It consists of the following:

 
Thanks for taking the time to post that, but avoiding measuring the distortion during the transient is kind of missing the point, isn't it? And the FFT combines all results during the sample period but doesn't show how it changes over time, which is what seems most important here.
 
Can you subtract a normalized output from the input, and create a difference plot? (magnified, and including transients)

 

[stv014]

  Thanks for taking the time to post that, but avoiding measuring the distortion during the transient is kind of missing the point, isn't it? And the FFT combines all results during the sample period but doesn't show how it changes over time, which is what seems most important here.

 
The FFT is not taken during the transients because then the spectrum would be dominated by the transient itself, which has a wide bandwidth, and the low level distortion products would be swamped.
 

Can you subtract a normalized output from the input, and create a difference plot? (magnified, and including transients)

 
That can be done easily, although note that without equalizing the undistorted signal, there will always be some difference because of the slight frequency response errors (which are not non-linear distortion).

 

[lamode]

   
The FFT is not taken during the transients because then the spectrum would be dominated by the transient itself, which has a wide bandwidth, and the low level distortion products would be swamped.
 
That can be done easily, although note that without equalizing the undistorted signal, there will always be some difference because of the slight frequency response errors (which are not non-linear distortion).

 
Perhaps you can apply a 20kHz low pass to the transient before running it through the test? That might be more representative of a musical transient.

 

[stv014]

Here are some difference waveforms for the above test. First the original output sample for reference:

Again, one sample is 100 ns, so the total duration shown on the picture is 5.5 s. The test signals are:
- DC level = 0.5, at 0.5 s and 2.5 s for 1 s
- 10 kHz sine at 0.25 level, at 2 s for 1 s (overlaps with the second DC step)
- 19 and 20 kHz sine at 0.475 level, at 4 s for 1 s
 
The full difference signal at ~2x and ~200x vertical zoom (left channel = raw difference, right channel = equalized difference):
   
The equalized difference was calculated by passing the original input signal through two -6 dB/octave lowpass filters (fco = 500000 Hz and 1000000 Hz) and a parametric equalizer (3.8 dB peak = 1.5488x gain centered at 707100 Hz, Q = 0.7071), and then subtracting this from the output of the simulated feedback loop. The filtering does not match the frequency response perfectly, so the difference is still not pure non-linear distortion, but it is a significant improvement over the raw difference, and the peak level is now reduced to about -48 dBFS.
 
This is the difference at the step during the 10 kHz tone (horizontally zoomed on the right):
   
And, for comparison, only the step without the 10 kHz tone:

The only obvious effect on the tone is the constant increased distortion after the DC offset is added.
 
Finally, the difference at the beginning of the 19+20 kHz IMD test:

As expected, it does not show any temporary increase in the distortion.